An Introduction to Mirages

Introduction

First of all, what's a mirage? Mirages are not optical illusions,
as many people (and Web sites!) think. They are real phenomena of
atmospheric optics, caused by strong
ray-bending
in layers with steep
thermal gradients.
Because mirages are real physical phenomena, they can be
photographed.

Optical
illusions,
on the other hand, are perceptual
quirks of human vision, in which the observer sees something that does not
exist physically. Of course, the distorted images produced by mirages may
elicit optical illusions, when an observer
misinterprets the scene —
hence, the confusion of these distinctly different classes of phenomena.
(For many examples of optical illusions, please see the Web pages of
Akiyoshi Kitaoka,
a perceptual psychologist in Japan.)

In a mirage, there is at least one inverted image of some object.
This “mirror image” is the
origin
of the French word mirage,
which comes from the phrase se mirer,
“to be reflected; to see one's image in a mirror.”

Often, a mirage contains multiple images, alternately erect and inverted.
Mirages are classified according to the number and relative positions of
these images. The classical mirages are:

Number of images

Name

Description

2

Inferior mirage

Inverted image below erect one

2

Superior mirage

Inverted image above erect one

3

3-image mirage

Inverted image between erect ones

many

Fata Morgana

complex alternation of distorted erect and inverted
images

In addition, there are the recently-recognized
“mock mirage” and
Alfred Wegener's
“late mirage” or Nachspiegelung. As these have a
different optical mechanism, I like to call them pseudomirages. Because
refraction
displaces images primarily in a vertical plane, the various
images are usually stacked up directly on top of one another.
The different
types
are described in a little more detail
here.

Mirages are distinguished from other
refraction phenomena
such as
looming
(visibility of distant objects usually hidden below the apparent horizon),
towering
(exaggerated vertical size of images),
sinking
(disappearance below the horizon of objects usually seen),
and
stooping
(images squashed together vertically),
in which an object may appear distorted, but not inverted. Some of these are
milder versions of the phenomena that produce mirages.

Common misconceptions

Location

It is incorrect to say (as even some textbooks do) that a mirage is an
image in the wrong place, because atmospheric refraction displaces almost
everything we see from its geometric position — that is, rays of light
in the lower atmosphere are usually
curved,
because the density of air
usually decreases steadily with increasing height. Thus, everything
normally appears displaced slightly above its geometric or
“true” position.
This displacement is known as terrestrial refraction when the
object is inside the atmosphere, and astronomical refraction when
it is beyond the atmosphere. While these effects are usually small
enough to escape casual observation with the naked eye, they are very
severe problems in fields such as geodesy and positional astronomy,
because they can be hundreds or even thousands of times larger than
measurement errors.

Reality

Sometimes people think that the erect image of a classical mirage is
“the object itself,” and that the inverted one is
“just a mirage,” and somehow not real.
But this notion is challenged by 3-image (and other multiple-image)
mirages, in which two or more erect images occur: which one, then, do you
choose as “the real one”?
In fact, all the images are equally genuine; every one of them is
as truly “the object itself” as any other,
including the inverted ones.
And as all the images are in general displaced from the geometric
position of the object, location is no indicator of legitimacy.

Indeed, from the point of view of geometrical optics, it is the inverted
images that are “real images.” And the erect images are merely
“virtual images” to an opticist —
including the ordinary appearance of
objects, even without mirage conditions! So it is best to recognize that
we are really seeing “the object itself” in every image, even when
there are many of them — while bearing in mind that
everything we see is slightly displaced
from its “true” or geometric position by atmospheric refraction,
whether there is a mirage or not.

Size

Another common misconception is that the miraged image can fill a large
part of the sky, as in
this
old drawing. Hogwash! Mirages NEVER look like that! They're
always confined to a narrow strip of sky — less than a
finger's width at arm's length — at the horizon.

Green flashes and mirages

Green flashes
are colored phenomena due to the
dispersion
of atmospheric refraction.
While every refraction phenomenon has some dispersion connected with it,
the dispersion is inappreciable under most circumstances.
However, certain mirages produce much larger dispersion effects; and the
most spectacular of these are the green flashes.
So, to understand green flashes, it is first necessary to understand
mirages.

The relation between solar and terrestrial mirages is beautifully
illustrated by the photographs at the right, taken by
Mila Zinkova
in San Francisco. The nine separate frames show the Sun setting behind
some terrestrial object,
which is seen silhouetted against the Sun in the first 5 or 6
frames. Several of the frames show multiple, highly flattened
images of the entire Sun.
(A larger version is available
here.)

The mirage of the foreground object, which resembles a chess piece,
also displays multiple images: notice the repetition of both wider and
narrower parts of it, at different heights.
You can see that a detached image of part of this object appears above the
main “chess piece” in the first 4 frames.
There is a general correspondence of
features in the solar and terrestrial mirages: the “chess
piece” is wider in strips of the solar image that appear darker.

Even though there's no green flash in this sunset, you can see that the
atmospheric structures that produce mirages of terrestrial objects also
distort the Sun's image. However, this example is so complex that
it's difficult to sort out all the details.

Mila has found
another
photograph that illustrates the correspondence between the distorted low
Sun and miraged topography. In addition to showing that both the Sun
and the mountains are so vertically stretched as to have nearly vertical
sides in the same zone of altitudes, that picture has a reflection of
the Sun on a pool of water in the foreground, and a “glitter path”
on the water beyond it, making a very pretty composition.

It's much easier to understand mirages after you have seen more of them.
Fortunately, the Finnish photographer Pekka Parviainen has made a
fine selection of his mirage photographs available on the Web.
Take look at them if you are unfamiliar with mirages.
Furthermore, Pekka now has
his own site, with even more mirage
pictures.

A “textbook example” of a superior mirage was taken by
Wim van Bochoven in late May, 2002. He has made his pictures available to me;
here they are,
with a description of the details.
His sequence shows many characteristic features very clearly.

A good collection of pictures showing Fata Morganas is available at Dave
Walsh's
website.
He has a good mixture of pictures showing the “striated zone” that
Forel
says is the characteristic feature of the Fata Morgana, and more usual
multiple-image superior mirages. Be sure to explore the links on his page
to the larger, higher-resolution versions of this Arctic-mirage slideshow.

Another site with some good images is
Olaf Squarra's pages.
He shows a fine multiple-image mirage of the Vestmannaeyjar Islands as
seen from Eyrarbakki, on the southern coast of Iceland.
There is also a Fata-Morgana type of mirage of farm buildings, shown at
the right side of his page; be sure to look at the enlargement.

Some very nice
multiple-image
mirages,
perhaps complex enough to count as
Fata Morgana displays, were taken at the Weather Service office in
Rapid City, South Dakota, at sunrise on Dec. 19, 2000. Evidently the
nocturnal radiative inversion is responsible for this display.

Mila Zinkova's
series of pictures
showing the variations in a superior-mirage display within a few minutes
is also very beautiful and instructive. It shows the alternations of
compressed and stretched zones in the image that characterize the
“Fata Morgana” type of display. She has another fine
example of this type
here,
as well as an
animation
of a superior mirage display (near the end of that last page).

Mila has also pointed out a
page
of excellent superior-mirage images taken by C. B. Clements, showing
mirages of ships at the horizon. The 3rd (uppermost) image of the 3-image
mirage is evident in some of these — particularly numbers 1, 7, and 8.
(This is one of those “album” pages, where you have to click
on the thumbnail images to see a larger version.)
If you don't want to go through all that, there's a nice big version of
one
of these on
Les Cowley's website.

Another fine set of mirage images, photographed through a telescope, is
shown on the
website
of an Australian radio operator, who uses the associated
ducts
for long-distance radio communication. (See his other pages for the radio
logs.) Sample his list of “Older Images” to see aditional
details.

Ctein has a
very fine inferior-mirage picture
that shows many features of these mirages very distinctly. The mirage
begins at a sharply defined boundary where the line of sight meets the
smooth desert floor at the critical angle; the mirage image is rather
irregular near this foreground boundary
(because the “reflection” occurs
so close to the ground surface at first that every little irregularity in
the surface produces a corresponding wiggle in the reflected image), but
with increasing distance behind the boundary, the image becomes smoother
(as the turning point of the rays moves upward in the air, and so becomes
less sensitive to small deviations from flatness in the ground). The
mirage begins close enough to the camera (only a kilometer or two away)
that there isn't much atmospheric scattering in between, so the colors in
the erect and inverted images are clear and distinct, and you can see how
the inverted images have the same color and brightness as the erect ones
— while the background is a bluish-gray because of intervening airlight.
The turbulence produced by the convection currents rising from the hot
ground surface makes many lines that should be straight in the picture
appear ragged and irregular. Nice.

Another
example
came from Corindi Beach on the eastern coast of Australia.
[The original link is no longer available, but the
Wayback Machine
provides a copy.]
The observer, David James, provided a link to a [now also archived]
map
that shows where the pictures were taken; he says he was standing at the
point where the arrow on the map meets the sea, about 7.2 meters above the
sea.
(Normal refraction would put his horizon about 10 or 11 km
away; however, with the stronger refraction indicated by the mirage, it
must be farther away.)
The small island that is clearly visible to the right of the mirage in his wide
angle shots is North West Solitary Island, about 7 or 8 km from the camera.
The mirage was of North West
rock, North Solitary Island and the dot on the map above North West Rock,
all 25 km or more away.
The pictures were taken between 6:11 and 7:47 a.m.; the temperature was
18°C, but rose to 33°C later in the day. (Probably this was also
about the temperature of the hot air above the inversion.) The day was
nearly calm — typical of conditions that produce strong inversions,
and superior mirages.
The date was 26 September 2003, which is early spring in the Southern
Hemisphere — again, typical season for mirages in temperate
latitudes.
The nearby islands that are seen undistorted lie a little inside the observer's
horizon; the miraged objects are beyond it.
Pretty much a textbook example, I'd say. Thanks for the details, David!

Mila Zinkova
has a couple of large images showing mirages of the Moss
Landing power plant from across Monterey Bay. The appearances on two different
days are quite different. Here's a link to
one,
and a link to the
other.
In both cases, the ducting has produced multiple images, mostly very
stretched vertically, below the strong inversion over the cold water.
The difference in appearance seems to be due mainly to a higher inversion
in the first example.

Interestingly enough, mirages seem to be common enough across Monterey Bay
that
another site
also discusses them, again featuring the Moss Landing power plant.
In addition, there's
yet another
mirage of the Moss Landing power plant on
Les Cowley's website.

For a good video of inferior mirages, take a look at Robert Krampf's
website,
which contains a short
video
of inferior mirages, both in the desert and on a highway.

Atmospheric refraction

Mirages are phenomena of atmospheric
refraction;
so to understand mirages, you first have to understand
refraction in the atmosphere.
(I've added a page that emphasizes some
basic principles
of atmospheric refraction; I hope it will serve as a sort of road map
that will keep you from getting lost in all the details.)

Because atmospheric refraction is ordinarily quite small — usually only a
fraction of a degree — these effects are not ordinarily visible to the
naked eye. However, they are easily measurable with optical instruments,
and so large that astronomers adjust the mountings of their telescopes
to minimize the inconvenient effects of atmospheric refraction.

The air near sea level is about a thousand times less dense than water;
but that's still enough to change the direction of light
rays that enter
it from a different medium, such as the nearly empty space outside our
atmosphere.
And, because the
density
of the air changes continuously with height
above the Earth, light rays within the Earth's atmosphere
bend
continuously as they pass from one level to another.
Usually, the density decreases steadily from the ground up, so the rays
(which always bend toward the denser material)
are concave toward the surface of the Earth.

Standard Refraction

Here's a picture of a few nearly-horizontal rays as they pass through the
lowest 9 kilometers of air. The label at the outer (right)
end of each ray gives its angular
altitude
above the horizontal (in
minutes of arc)
at the observer's eye, which is at the vertex of the fan of rays,
near the base of the scale of heights at the left.
(The diagram is compressed horizontally to
magnify small angles above and below the horizontal; “Hobs” denotes the
height of the observer.)

You can see that the ray marked with a zero, which is horizontal at the
observer's eye, is bent down appreciably below the horizontal where it
passes the 9 km height above the Earth's surface. Even the ray that the
observer sees at 20 minutes of arc above the horizontal is bent so that
at 9 km, it's coming from a direction a little below the
astronomical horizon.
The rays are traced exactly
for the Standard Atmosphere,
for an observer 12 meters (about 40 feet)
above sea level.

The diagram above shows only the bottom 2/3 or so of the atmosphere, so it
depicts only 2/3 of the total bending, which is about 34 minutes (just
over half a degree) for a horizontal ray.
Even though the vertical exaggeration of this diagram is about a factor of
about 13 (compare the vertical and horizontal scales), the
ray curvature
and refraction are barely visible.

To make these effects more apparent, let's increase the vertical
stretching, and show just the lowest few meters of air. Here are some
rays traced for the same observer, 12 meters above the surface of the
Earth, but now only the lowest 24 meters of the atmosphere are shown. The
vertical exaggeration is now about 250, and you can see the ray
curvature better. (If the curvature isn't obvious, try holding a straight
edge up to the longer rays.)

Notice that because the observer is 12 meters above the Earth's surface,
sky as much as 6 minutes of arc below the horizontal can be seen; the
apparent horizon
(marked by the ray that just touches the Earth's surface)
is 6 minutes below the astronomical one.
This depression is called the
dip
of the horizon.
That lowest line of sight just grazes the surface of the Earth some 13
or 14 km from the observer, so the distance to this observer's
apparent horizon is nearly 14 km, under standard conditions.
Because of refraction, the apparent horizon is a little farther away, and
has less dip, than one would expect from
a straight line drawn from the eye tangent to the Earth, ignoring
refraction.
(See the
distance to the horizon page
for more discussion of this matter.)

Because refraction makes the rays concave toward the Earth, objects near
the horizon appear at slightly higher
altitudes
than their geometric
positions (i.e., where they would appear if there were no refraction).
This is true of both objects on the Earth (where the displacement is
called terrestrial refraction)
and objects in the sky (astronomical refraction).
(Additional ray-traces and simulations of images produced by
terrestrial refraction in the Standard Atmosphere are shown
here.)

And because the astronomical refraction, unlike the
terrestrial
refraction, increases rapidly with decreasing
altitude, the lower limb of the Sun or Moon is raised up more than the
upper one — so these objects appear slightly
flattened
near the horizon.
You can see an example of this flattening produced by the Standard
Atmosphere in the
simulation
of a “textbook” sunset.

So far, we've considered only the Standard Atmosphere
— a convenient fiction that is far from typical conditions, even though
it is close to the average of a large number of measurements.
The real atmosphere is never so featureless; all the complexities of
reality have been averaged out here.
Ordinarily, the real atmosphere has an irregular temperature profile, and
the irregularities produce much more interesting refraction phenomena than
we have seen so far.

Although air is a thousand times less dense than water, and its
refractivity
is about a thousand times smaller, atmospheric refraction can produce effects
that are sometimes easy to see, or even spectacular. Mirages and green
flashes are the most striking of these effects.

What's a Mirage?

Mirages are multiple images formed by atmospheric refraction.
(Other refraction phenomena
include the ever-present small vertical
displacements of everything we see — called terrestrial or
astronomical refraction, as the displaced objects are on the Earth or in
the sky — and the
less common phenomena
of looming and towering, or
stooping and sinking.)
I favor the usage established by
Pernter and Exner
in their
classic book,
Meteorologische Optik :
refraction phenomena include such
image distortions as looming and stooping, as well as everyday terrestrial
and astronomical refraction;
but “mirage” is restricted to
displays in which two or more images of the same object appear.
A similar division is used in
W. J. Humphreys's bookPhysics of the Air.
Another excellent discussion (in French) was published by
Bonnelance (1929).

An important theorem
of atmospheric optics tells us that multiple or inverted images of
celestial objects can only occur below the astronomical horizon.
Thus, the height of the eye, which determines the dip of the apparent
horizon, and thus the width of the zone of sky between it and the
astronomical horizon, is very important for understanding mirages.

Also, if an object is to appear inverted, rays from the top of the object
must cross rays from the bottom of the object on their way to the
observer's eye, so that the former arrive at the eye below the latter.
Watch for such ray-crossings in the diagrams shown below, and in the
additional
mirage simulations.

There are several different
kinds
of mirage. Each involves a
particular thermal structure in the atmosphere, and a particular relative
placement of the structure and the observer.
Here are some common examples:

The Inferior Mirage

The commonest mirage is the “inferior” mirage —
so called not because it is a poor example, but because the inverted
image lies below the erect one. This is sometimes called the
mirage of the desert, but today it is more appropriately known as the
“hot-road” mirage, as it is seen on asphalt paving nearly
every sunny day.

The association with deserts is wildly inappropriate,
as the name mirage was originally French sailors' jargon.
Although these mirages are readily seen both at sea and on sand, they are
really typical of smooth surfaces, and can be seen over ice and
snow as well.

In this mirage, the surface of the Earth, heated by the Sun, produces a
layer of hot air of lower density just at the surface.
Grazing rays bend back up into the denser air above:

Here an observer 5 m above the surface sees an apparent horizon with a dip
of 6 minutes of arc, just as did the observer at 12 m in the Standard
Atmosphere. But the apparent horizon is now only 4 km away; the circle of
the horizon has apparently shrunk.

If you place a vertical ruler between the observer and the apparent
horizon, you will see that objects closer than the apparent horizon are
nearly undistorted: the rays are about equally spaced at the object (the
ruler).
But beyond the horizon, the object is strongly distorted. For example, at
10 km, some parts of the object near the ground are invisible, and other
parts are seen twice: the point where the rays marked
−2 and −6 cross is seen at both these altitudes.
Also, a small interval of the object appears to the observer to fill the
interval from −3 to −5 minutes of altitude, and so is
strongly magnified vertically. The whole interval of
sky from −4 to −6 minutes appears upside down to the observer.

As a vertical object recedes beyond the apparent horizon at 4 km distance,
more and more of its lower parts disappear; and more and more of its upper
parts are imaged twice, with the inverted image below the erect one.
Note that, because of the curvature of the rays, even the erect image is
displaced upward from its geometric direction from the observer.

Thus, for example, the writer and botanist
Adelbert von Chamisso
described a fine inferior mirage as follows:

I saw a surface of water before me in which a low hill was reflected
that extended along the opposite shore. I went toward this water. It
disappeared before me, and I reached the hill with dry feet. When I had
covered about half the distance, I seemed to Eschscholtz, who had remained
behind, to have been submerged up to my neck in the reflecting layer of
air, and, shortened the way I was, he said I looked more like a dog than a
human being. As I strode onward, toward the hill, I emerged more and more
from the layer of air, and I appeared to him, lengthened by my reflection,
to get taller and taller, gigantic, slender.

When land rises above the horizon, as seamen are wont to express it,
the line that is taken to be the horizon is the edge of a reflecting
surface formed by the lower layer of air and closer to the eye; a line
that really lies below the visible horizon. I believe that this illusion
in some cases can have an influence on astronomical observations and can
cause an error in these of five and perhaps more minutes.

The Superior Mirage

The textbook writers are fond of saying that the superior mirage is just
like the inferior mirage, but upside down; that is, the “reflecting”
layer in the atmosphere is overhead instead of underfoot.
But that statement isn't quite right, because the curvature of the Earth
can't be inverted, and it plays an important part in these phenomena.
Furthermore, the inferior mirage is somewhat simplified because rays can't
pass through the Earth.
But when the “reflecting” layer is overhead, additional phenomena can
occur because rays can penetrate far beyond it.

Here's a ray diagram for a superior mirage.
To make the “reflection,” I've put a little
zigzag
into the temperature profile of the Standard Atmosphere: there's a
thermal inversion
of 2°C between 50 and 60 m height above the surface.
(This inversion layer is shaded in the diagram.)
The observer is placed 54 m above the surface, just below the middle
of the inversion.

The ray curvature within the inversion is stronger than the curve of the
Earth, which produces some remarkable effects.
Before considering the details, let's look at the general features.

There are really three different groups of rays, corresponding to
three parts of the observer's sky.
First, the rays well above the horizon are refracted more than usual —
especially the lower ones (notice how the rays spread farther and farther
apart as you go from 10 to 8 to 6 to 4 minutes of arc above the
astronomical horizon).

Then, as the eye looks lower in the sky, there is a sudden transition to
rays that are
trapped
beneath the inversion.
The rays marked 2, 0 and −2 at the lower right corner of the
diagram never rise above the top of the inversion, but are repeatedly
turned back down within it.
The region from 45.3 to 60 meters above the surface forms a
“duct” that
guides these rays around the curve of the Earth (in principle, forever; in
the real world, the strong inversions that produce ducts have limited
horizontal extent, so the rays eventually escape.)

Because so many rays overlap in the duct, this region is pretty cluttered
in the original diagram. So here's a closer look at just those three ducted
rays:

You can see that these three rays just oscillate back and forth in a
limited interval of height, never reaching the top of the inversion layer.
If these rays don't get out of the duct, they certainly don't get out of
the atmosphere, so the Sun and Moon can't be seen through this
band of sky
parallel to (and centered on) the
astronomical horizon.

Notice, too, that these rays have their lowest points below the
inversion. So the duct is deeper than just the inversion itself.
And finally, notice that these trapped rays are always concave toward the
Earth, even at the bottom of the duct: as
Wegener
emphasized, the
curvature of the Earth makes this possible, as it bends down faster than
the rays themselves do below the inversion.

A careful look at the diagram shows that the three rays all cross about 20
km from the observer. Objects beyond this distance — say,
30 km
from the eye — are seen inverted.
This inverted image is the classical “superior mirage,” in which the
inverted image appears above an erect image (formed by a third group of
rays that will be discussed below).

But the rays are “reflected” again, and cross
again about 40 km from the observer, so that at
50 km
an erect image is seen once more.
Notice, however, that the crossings are gradually
getting out of step,
so that the image becomes progressively more distorted and
eventually quite jumbled.
This jumbling leads to the confused images of the Fata Morgana,
a complex mirage display produced by strong ducts.

Wegener's Nachspiegelung or
“late mirage”

The third group of rays lies below the duct, as seen by the observer.
These are the rays marked with negative altitudes, from −4 to
−10 minutes.
Because they make considerable angles with the horizontal within the duct,
they are not trapped, but are merely bent strongly within the inversion
layer, and extend beyond it.
There is a considerable angular gap
at the top of the atmosphere between the first group of rays
(above the duct) and the last group (which pass below the duct and back up
through it, looking to the right in the diagram).

Here's a cleaner view of these rays that cross the duct.

Notice that the rays that the observer sees at altitudes of −4 and −6
minutes of arc actually cross, about 50 km away.
So objects more than 50 km from the observer, and 100 m or so high, can be
seen miraged (inverted).
But there usually isn't anything but air for the observer to see in this
region, so these mirages are not often noticed.
However, beyond the atmosphere, there may be the
setting Sun
in this part of the sky; then it is miraged in what
Alfred Wegener
called the
Nachspiegelung,
or “late mirage” that appears after the Sun has
“set” on the upper side of the duct and then reappeared below it.
There are some examples of this mirage on the page devoted to
simulations
of ducted sunsets.

This is our first example of what may be called
a “pseudomirage” — a purely refractive
image inversion. There is no way the ray crossing shown here can be
regarded as due to a “reflection”, like the ordinary inferior and
superior mirages.
(Note that the French
word mirage that English has adopted is derived from the
phrase se mirer, meaning
“to be reflected; to see one's own reflection.”
This term was originally used by French sailors, and rather
reluctantly accepted in scientific usage around 1800.)

One way to explain this mirage is to regard the denser air below the
thermal inversion as acting like a simple convex lens, with a focal length
of about 12 km in the example here.
The mirage is then the inverted real image of distant objects that such a lens
produces.

There are other kinds of pseudomirage, all (so far as I know) associated
with thermal inversions — though not all require ducts.
In fact, Wegener's late mirage is possible, though weak, with an inversion
layer too feeble to produce a duct.
Thus the superior mirage always produces a late mirage, but the reverse is
not true.

The mock mirage

Now suppose the observer is above the inversion.
We get a ray diagram that looks like this:

Once again we have ray-crossing (look at the ray that was at −4 minutes'
altitude at the observer:
it is well below the three rays at −6, −8, and
−10 minutes, and crosses all of them); so we have inverted images.
This strongly-depressed ray enters the top of the inversion near grazing
incidence, so it has the longest path in the inversion of any ray shown
here. But the inversion is where strong bending occurs; so this ray is
bent down the most. The others pass through the inversion more
obliquely, and so have shorter paths in it, and less bending.

As in Wegener's “late mirage,” this is a purely refractive image
inversion. In fact, a similar explanation holds here: the dense air below
the inversion acts like a lens, forming a real, inverted image of distant
objects seen through it. So this is another example of a pseudomirage.
We have called it
a “mock mirage.” Like Wegener's late mirage, it really
requires only an inversion, not actual ducting.

Again, only distant objects above the base of the inversion can be seen
miraged, so the mock mirage is rarely seen with terrestrial objects.
(To see what terrestrial mock mirages can look like, see their
simulations.)
But it is a
common cause of low-Sun distortion, and a common kind of green flash.
Such a green flash is shown in the
sunset simulations.

Still other mirages and green flashes can occur.
But the ones shown above are certainly the most common.

Reflections or Refractions?

Pedants and purists insist that all mirages are purely refraction
phenomena; and, strictly speaking, they are correct.
Nevertheless, it is so useful to regard the classical inferior and
superior mirages as
approximately due to internal reflection that we cannot avoid
considering that point of view — but with the caveat that
certain subtle phenomena,
which turn out to be essential for green flashes, are not
explained by the “reflection” approach.

Furthermore, one should bear in mind that the words for
“mirage” in the most common European
languages are all derived from terms
meaning “reflection.” I have already pointed out that the
French term “mirage” is derived from a phrase meaning
“to be reflected.” In German and Dutch, the terms used
translate literally to “air-reflection.”
Thus the notion of reflection is implicit in every discussion of mirages
— though
Everett's
term “quasi reflection” is closer to the truth.
I think this close association of “reflection” with the classical
inferior and superior mirages is a good reason to distinguish them from
the “pseudomirages,” which are purely refractive.

Still, one can regard
all these inverted images as real images (in one dimension) formed by
a positive lens. The finite focal length of the lens explains why there
is a minimum
distance
to objects that are seen miraged. From this point of view, even the
classical mirages are refraction, rather than reflection, phenomena.

Green Flashes

Now it begins to be possible to explain some common green flashes. The
commonest is certainly the
inferior-mirage
flash.
At the transition between the erect and inverted images of the inferior
mirage, there is a zone of sky, parallel to the horizon, in which strong
vertical stretching occurs. This broadens the Sun's upper green rim,
which is normally too narrow to be seen without optical aid, into a
feature wide enough to see easily. I have made some
simulations
that illustrate both
the narrowness of the
green rim
in standard conditions, and its
exaggeration into a
green flash
by an inferior mirage.

Although many textbooks treat only what I might call the “classical”
mirages — namely, the inferior and superior mirages — which are
roughly describable in terms of total internal reflection of rays incident on a
less-dense layer of air from a denser one, green flashes show that other
mirage-like phenomena are equally important.
Many of these (such as the mock mirage) are pseudomirages.

The commonest of the pseudomirage flashes (namely, the
mock-mirage flash)
is
illustrated
in the simulations.
Unfortunately, this turned out to be a more complicated problem than I
had thought: the width of most mock-mirage flashes is
due to the transitions
between thermal inversions and the adjacent layers of
air, and this structure presents difficult numerical problems — roundoff
errors due to subtraction of nearly-equal quantities — that have been
tricky and difficult to overcome.
So only in late 2002 was it possible to present reasonably accurate
simulations of the mirages and green flashes associated with thermal
inversions, and particularly, with ducting.
And the phenomena produced by ducts are so complicated that I have made a
special page
to show their effects on sunsets.

(Eventually, I plan to put up some additional pages that add the effects of
dispersion to the mirage theory, and thus show in detail how the green
flashes that accompany the various mirages are produced.
That will require some additional software to draw the necessary diagrams.)

Boundary Layers and Mirages

Where do the thermal structures that cause mirages come from?
They aren't in the
Standard Atmosphere.

Air is generally denser near the surface of the Earth, where it is
compressed by the weight of the air above it, and less dense higher up.
Nevertheless, the Earth's surface is often appreciably warmer or cooler
than the overlying air; then there is a relatively thin
“boundary layer”
in which the transition between the conditions at the surface and those in
the free atmosphere takes place.
The depth of this transition may vary from a few millimeters to a few
hundred meters.
(There is a whole field called “boundary-layer meteorology”
that deals with this layer; a good introduction is available at a
website
devoted to hot-air ballooning.)

There can also be “internal boundary layers” between large bodies of
warmer and colder air.
A good example is the cold pool of air that fills valleys at night:
between this (which is produced by the nocturnal boundary layer of cold,
dense air that slips down the hillsides as they cool by radiation to
space) and the overlying warm air left over from the day before, there is a
relatively sharp transition.

Other sharp thermal boundaries — especially, strong inversions
— are
associated with weather fronts. Still others are produced by turbulent
mixing, where there is wind shear. The result is that the atmosphere is
full of thermal structures that are omitted from simple “textbook”
idealizations like the
Standard Atmosphere.

The feature of these boundary layers that is significant for atmospheric
optics is their steep temperature gradient.
It is not uncommon to find changes of temperature of several degrees in a
meter, or less.
Because the density of air depends on temperature as well as pressure,
these boundary layers are regions of steep changes in density — and
hence, refractivity.

Thus, these strong changes of atmospheric density are nearly always
present. That means that mirages of some kind are nearly always present
in the atmosphere, if you put your eye at the right height.
Generally, the strongest optical effects require that the observer be close
to the height of the thermal structures.
Often, only the air itself is miraged, and the mirage is invisible (unless
there are clouds at the right height to appear distorted).
So, although mirages are nearly always possible, they aren't always
visible; and many are so close to the limit of resolution of the eye that
they usually escape notice. But an attentive person — especially, one
armed with binoculars — can see these phenomena remarkably often.

Mirages and sound propagation

A reader of these pages has asked me about possible relations between
mirages and sound propagation: in effect, “are there acoustic as well as
optical mirages?”
The answer is, “Sort of.” As the speed of sound depends on the air
temperature, the thermal structures responsible for optical mirages also
can produce peculiar sound propagation.

However, the speed of sound is essentially independent of the air density
(and hence, pressure); so the refraction of sound waves is quantitatively
different from that of light waves. Also, the refraction of sound waves
is strongly affected by wind shear, which is completely unimportant for
light refraction. So, although there are similar phenomena in acoustics,
you generally don't expect the acoustic “images” to resemble
the optical
ones. In particular, you can't expect to hear sounds coming from (optically)
miraged objects.

There's a lot of interesting information about sound propagation and
thermal inversions in the atmosphere at
Mike O'Connor's page
on this topic. You can also get there from
his main page
by clicking on the “Information” item in the menu bar at the top.

More information

If you want to know more about mirages, I have a few recommended mirage
references on the reading page.
There are additional ray diagrams for
ducts
on the
duct page.
And some
simulations
are available for the commoner types of mirages.

If you found my use of italics and boldface in links
confusing, please read the page
explaining this typographic convention.